<s>
The	O
spectral	B-Algorithm
concentration	I-Algorithm
problem	I-Algorithm
in	O
Fourier	O
analysis	O
refers	O
to	O
finding	O
a	O
time	O
sequence	O
of	O
a	O
given	O
length	O
whose	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
maximally	O
localized	O
on	O
a	O
given	O
frequency	O
interval	O
,	O
as	O
measured	O
by	O
the	O
spectral	O
concentration	O
.	O
</s>
<s>
For	O
a	O
given	O
frequency	O
W	O
such	O
that	O
0	O
<	O
W	O
<	O
½	O
,	O
the	O
spectral	O
concentration	O
of	O
U(f )	O
on	O
the	O
interval	O
 [ -W , W ] 	O
is	O
defined	O
as	O
the	O
ratio	O
of	O
power	O
of	O
U(f )	O
contained	O
in	O
the	O
frequency	B-Algorithm
band	I-Algorithm
 [ -W , W ] 	O
to	O
the	O
power	O
of	O
U(f )	O
contained	O
in	O
the	O
entire	O
frequency	B-Algorithm
band	I-Algorithm
 [ -½ , ½ ] 	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
is	O
there	O
a	O
sequence	O
for	O
which	O
the	O
sidelobe	O
energy	O
outside	O
a	O
frequency	B-Algorithm
band	I-Algorithm
 [ -W , W ] 	O
is	O
minimum	O
?	O
</s>
<s>
this	O
matrix	B-Architecture
lie	O
between	O
0	O
and	O
1	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
number	O
of	O
dominant	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
M	O
that	O
are	O
close	O
to	O
1	O
,	O
corresponds	O
to	O
N	O
=	O
2WT	O
called	O
as	O
Shannon	O
number	O
.	O
</s>
<s>
Not	O
limited	O
to	O
time	O
series	O
,	O
the	O
spectral	B-Algorithm
concentration	I-Algorithm
problem	I-Algorithm
can	O
be	O
reformulated	O
to	O
apply	O
on	O
the	O
surface	O
of	O
the	O
sphere	O
by	O
using	O
spherical	O
harmonics	O
,	O
for	O
applications	O
in	O
geophysics	O
and	O
cosmology	O
among	O
others	O
.	O
</s>
